Creating the Mandelbrot set with R

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About the Mandelbrot set:

See here: https://en.wikipedia.org/wiki/Mandelbrot_set

Code

knitr::opts_chunk$set(echo = TRUE)
library(caTools) # external package providing write.gif function

jet.colors <- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F",
                                 "yellow", "#FF7F00", "red", "#7F0000"))
dx <- 400                  # define width
dy <- 400                  # define height
C <- complex( real=rep(seq(-2.2, 1.0, length.out=dx), each=dy ),
              imag=rep(seq(-1.2, 1.2, length.out=dy), dx ) )
C <- matrix(C,dy,dx)       # reshape as square matrix of complex numbers


Z <- list(0,0,0,0,0)                     # initialize Z to zero

# initialize output 3D array
X<- list(array(0, c(dy,dx,100)),
     array(0, c(dy,dx,100)),
     array(0, c(dy,dx,100)),
     array(0, c(dy,dx,100)),
     array(0, c(dy,dx,100)))

for (k in 1:100) {          # loop with 20 iterations
  for (i in 1:5){
  twoI <- 2*i
  Z[[i]]<- Z[[i]]^(twoI) + C               # the central difference equation
  
  X[[i]][,,k] <- exp(-abs(Z[[i]]))   # capture results
  }
}

for (i in 1:5){
    filename <- paste("Mandelbrot",as.character(i),"a.gif",sep = "")
    write.gif(X[[i]], filename, col=jet.colors, delay=100)
}

Results

Mandelbrot set ( \(z = z^2 + c\))

Another fractal (\(z = z^4 + c\))

Another fractal (\(z = z^6 + c\))

Another fractal (\(z = z^8 + c\))

Another fractal (\(z = z^{10} + c\))

Sources

• Whoever wrote this cool example: https://en.wikipedia.org/wiki/R_(programming_language)#Programming_features
• How to compile a Rmarkdown notebook to html: http://kbroman.org/knitr_knutshell/pages/Rmarkdown.html